Method for estimating the characteristic parameters of a cryogenic tank, in particular the geometric parameters of the tank

ABSTRACT

(EN) The invention relates to a method for estimating the characteristic parameters of a cryogenic tank ( 1 ), in particular geometric parameters, including: a step comprising the measurement of the pressure differential between the upper and lower parts of the tank prior to filling DP mes     —     before ; a step comprising the measurement of the pressure differential between the upper and lower parts of the tank after filling DP mes     —     after ; a step comprising the determination of the mass of liquid delivered (m delivered ) during filling; and a step comprising the calculation of a first geometric parameter (R) of the tank, namely the radius (R) which is calculated from equation (I), wherein g is the Earth&#39;s gravitational acceleration and MAVO is a density coefficient that is a function of the density of the liquid and the gas in the tank and optionally in the pressure measuring pipes ( 11 ) when the pressure differential is measured by at least one remote pressure sensor connected to the upper and lower parts of the tank via respective measuring tubes ( 11 ).

The present invention relates to a method for estimating characteristicparameters of a cryogenic tank and in particular geometric parameters ofthe tank.

The invention in particular makes it possible to improve the levelmeasurement in cryogenic tanks in order to improve the efficiency of thelogistic supply chain for supplying these tanks with liquid. The tanksconcerned comprise an internal fluid-storage tank (or internal barrel)placed inside an external tank (or outer barrel). These two barrels areseparated by a layer of insulation. The tanks store cryogenic liquidssuch as oxygen, argon, nitrogen with capacities of 100 liters to 100 000liters, for example. The storage pressures may range between 3 bar and35 bar.

The geometric parameters of an (internal) tank are needed notably inorder notably to estimate the level of liquid and the quantity that canbe delivered into the tank. Among the useful parameters, mention maynotably be made (in the case of a cylindrical tank with elliptical ends)of: the radius R, the total height of the tank (h_(tot)), the height Fof the elliptical part (end), the maximum height of liquid H_(max). Formany cryogenic tanks, these parameters are unknown or can be identifiedonly at the expense of significant work.

It is an object of the present invention to alleviate all or some of theabovementioned disadvantages of the prior art.

To this end, the method according to the invention, in other respects inaccordance with the generic definition given thereof in the abovepreamble, is essentially characterized in that it comprises a step ofcalculating a first geometric parameter (R) of the tank as a functionof:

-   -   a mass of liquid delivered (m_(delivered)) (in kg) determined        during the filling,    -   the difference between the pressure differentials (DP_(mes)) (in        Pa) measured before and after filling (DP_(mes) _(—)        _(after)−DP_(mes) _(—) _(before)), each pressure differential        measuring the pressure differential between the top and bottom        parts of the tank,    -   the densities of the gas and of the liquid (ρ_(g), ρ_(l)) in the        tank (in kg/m³).

Unless stated otherwise, the physical parameters are expressed in SIunits: distances (notably heights, radii, etc.) are expressed in meters(m), densities in kg/m³, volumes in m³, pressures or pressuredifferentials in Pa.

Furthermore, some embodiments of the invention may comprise one or moreof the following features:

-   -   the tank comprises a cylindrical portion and at least one end        having an elliptical portion of set height (F) (in m), and in        that the first geometric parameter is the radius (R) (in m) of        the cylinder,    -   the radius (R) is calculated using an equation of the type:

$R = \sqrt{\frac{m_{delivered}{g\left\lbrack {1 - ({MAVO})} \right\rbrack}}{\pi \left( {{DP}_{{mes}\_ {after}} - {DP}_{{mes}\_ {before}}} \right)}}$

in which g is the acceleration due to gravity of the Earth, in m/s² andMAVO is a dimensionless corrective coefficient that is a function of thedensity of the liquid and of the gas in the tank and possibly in thepressure measurement pipework when the pressure differential is measuredby at least one remote pressure sensor connected to the top and bottomparts of the tank via respective measurement pipes, and π is the numberPi

-   -   in order to estimate the density coefficient (MAVO), the method        uses at least one of the following assumptions:    -   the density of the liquid in the tank ρ_(l) is considered to be        equal to the mean of the densities of the liquid before ρ_(l)        _(—) _(before) and after ρ_(l) _(—) _(after) filling,    -   the density of the liquid before filling ρ_(l) _(—) _(before) is        equal to the density at equilibrium at the pressure of the tank,    -   the density of the liquid after filling ρ_(l) _(—) _(after) is        equal to the mean of, on the one hand, the density of the liquid        at equilibrium at the pressure of the tank before filling        weighted by the fraction of the volume occupied by this liquid        and, on the other hand, the density of the liquid in the truck        considered at equilibrium at the pressure of the truck, weighted        by the fraction of the volume available in the tank before        filling,    -   the volume of the liquid in the tank before filling is        considered to be equal to a known fraction (for example 30%) of        the maximum volume of liquid in the tank,    -   the density of the gas in the tank ρ_(g) is calculated at the        pressure of the tank P_(tank) and for a temperature that is        increased over the equilibrium temperature at the pressure of        the tank (for example increased by 20 K),    -   the measured pressure differential DP_(mes) is corrected to take        account of an additional pressure difference value (DP_(pipe))        created by the gas present in the measurement pipes in the event        of remote measurement,    -   the volume of the liquid V_(l) in a cylindrical tank of radius R        having one elliptical end of height F is given by the        relationship:

$\left. {{{If}\mspace{14mu} h_{l}} \geq F}\Rightarrow V_{l} \right. = {\pi \; {R^{2}\left\lbrack {h_{l} - \frac{F}{3}} \right\rbrack}}$$\left. {{{If}\mspace{14mu} h_{l}} < F}\Rightarrow V_{l} \right. = {{\frac{2}{3}\pi \; {FR}^{2}} - {{\pi \left( {F - h_{l}} \right)}\left\lbrack {R^{2} - {\frac{R^{2}}{3F^{2}}\left( {F - h_{l}} \right)^{2}}} \right\rbrack}}$

-   -   h_(l) being the height of liquid in the tank,    -   the method comprises a step of calculating a second geometric        parameter consisting of the height (F) of the elliptical portion        of the tank as a function of calculated value of radius (R), the        value of the height (F) of the elliptical portion being given by        the following equation:

$\frac{R}{F} = K$

-   -   K being a known dimensionless constant representative of a type        of tank or an arbitrarily chosen constant such as about 1.95,        -   the method comprises a step of calculating a third geometric            parameter consisting of the total height of the tank            (h_(tot)) from the pressure differential (in Pa) measured            just after a filling DP_(mes) _(—) _(after) assuming that            filling is total, this measured pressure differential being            expressed in the following form:

DP _(mes) _(—) _(after) =A ₀ H _(max) +A ₁ h _(tot) +A ₂ h _(g) _(—)_(before) +A ₃ h _(l) _(—) _(before)

in which A₀, A₁, A₂, A₃ are coefficients (in Pa/m) dependent on thedensities of the gas and of the liquid before and after filling, H_(max)is the maximum height of liquid in the tank, h_(g) _(—) _(before) beingthe height of gas in the tank before filling, h_(l) _(—) _(before) beingthe height of liquid in the tank before filling, and using the followingassumption: the height of liquid in the tank before filling h_(l) _(—)_(before) is estimated at a known set threshold FS expressed as apercentage of the maximum height of liquid H_(max), the height h_(g)_(—) _(before) of gas before filling being deduced therefrom as beingthe complement:

h_(l) _(—) _(before)=FS H_(max)

h_(g) _(—) _(before)=h_(tot−FS H) _(max)

-   -   the method comprises a step of calculating a fourth geometric        parameter consisting of the maximum height of liquid H_(max),        this being deduced the total height of the tank calculated from        the following equation:

$H_{\max} = {\frac{F}{3} + {\frac{V_{\max}}{V_{tot}}\left\lbrack {h_{tot} - \frac{2F}{3}} \right\rbrack}}$

-   -   the method comprises a step of calculating a fifth geometric        parameter consisting of the thermal loss of the tank, said        thermal loss expressed as an oxygen percentage (% O₂) of oxygen        lost per day being approximated using a relationship of the        type:

% O₂ lost per day=p_(l)V_(tot) ^(−p) ²

-   -   in which V_(tot) is the total volume of the tank, p₁ is a        coefficient of the order of 0.6 and preferably equal to 0.65273        and p₂ is a coefficient of the order of 0.3 and preferably equal        to 0.37149,    -   the method comprises a step of collecting a plurality of values        for the mass of liquid delivered (m_(delivered)) which are        determined respectively during a plurality of fillings of the        tank, and a step of calculating, for each value of mass of        liquid delivered (m_(delivered)), the fill ratio r

$r = {\frac{m_{delivered}}{{DP}_{mes\_ after} - {DP}_{mes\_ before}} = {{constant}\mspace{14mu} \left( {{in}\mspace{14mu} {{kg}/{Pa}}} \right)}}$

and the method of estimating uses only those values of mass of liquiddelivered (m_(delivered)) for which the absolute value of the fill ratior differs with respect to a set constant reference value by no more thana fixed threshold amount,

-   -   the set constant reference value for the fill ratio r consists        of the mean of this fill ratio r calculated for a plurality of        fillings.

Other specific features and advantages will become apparent from readingthe following description given with reference to the figures in which:

FIG. 1 depicts a schematic view illustrating a first example of acryogenic tank for implementing the invention (pipework outside thewalls of the tank),

FIG. 2 depicts a schematic view illustrating a second example of acryogenic tank for implementing the invention (pipework inside the wallsof the tank).

The method that is to be described hereinafter can be implemented by acomputer of a (local or remote) tank control system. This methodinvolves measuring a pressure and a pressure difference DP_(mes) and maycomprise remote transmission of data. The pressures are measured viapipework 11, 12 which may lie in the space between the walls of the tank(FIG. 2) or on the outside 11 (FIG. 1).

The tank 1 may comprise a pressurizing device such as a vaporizationheater 3 able to tap off liquid, vaporize it, and reinject it into thetank. This heater 3 regulates the pressure within the tank 1 in theconventional way.

For simplicity, the interior tank which stores the fluid willhereinafter simply be termed the “tank”.

The liquid supplied by a delivery truck during filling operations mayalso be considered to be in the state of equilibrium (temperature rangeof 10 K around equilibrium, for example 77.2 to 87.9 K in the case ofnitrogen). The pressure of the liquid in the delivery truck is chosen,according to the pressure of the tank, to be between 1 and 2 bar. Theliquid is introduced into the tank by pumping it.

Between two filling operations the tank 1 is subjected to the followingphenomena:

-   -   the quantity of liquid decreases (is consumed by the user), and        the corresponding drop in pressure is corrected by the heater 3,    -   heat enters the tank through the walls of the tank and the        heater (conduction, radiation).

After a certain time at equilibrium, liquid vaporizes in the tank andthis contributes to a loss of liquid. In addition, the density of theliquid decreases as the liquid heats up and, as a result, the liquidlevel is higher than if it had maintained its delivery temperature.

According to an advantageous particular feature, temperatures specificto the gas and to the liquid in the tank are considered, but withoutthese temperatures being a function of the location within the tank.What that means to say is that in what follows, the temperatures of thegas T_(g) and of the liquid T_(l) are mean temperatures.

The estimated liquid level is based on the pressure differentialDP_(mes) measured between the bottom and top ends of the tank.

According to the present method, the calculated height of liquid h_(l1)is calculated (in Pa) according to the formula (equation 1):

$h_{l\; 1} = \frac{{DP}_{mes}}{\rho_{l\; 1}g}$

Where ρ_(l1) is a calibration liquid density value (in kg/m³) that isconstant (but can be modified by an operator); g being the accelerationdue to gravity of the Earth in m/s².

Because the tank is not a geometrically perfect cylinder (its ends areelliptical, cf. FIGS. 1 and 2), the volume V_(l) of liquid uses twoequations according to whether the liquid level is below or above theelliptical part F (equations 2):

If h_(l1) is above the elliptical zone F

$\left. {then}\Rightarrow V_{l} \right. = {\pi \; {R^{2}\left\lbrack {h_{l1} - \frac{F}{3}} \right\rbrack}}$$\left. {else}\Rightarrow V_{l} \right. = {{\frac{2}{3}\pi \; {FR}^{2}} - {{\pi \left( {F - h_{l\; 1}} \right)}\left\lbrack {R_{2} - {\frac{R^{2}}{3F^{2}}\left( {F - h_{l\; 1}} \right)^{2}}} \right\rbrack}}$

R being the radius (in m) of the tank (in its cylindrical portion).

The mass of liquid contained in the tank m_(l) is deduced using thedensity of the liquid ρ_(l1) (equation 3):

m_(l)=ρ_(l1)V_(l)

The mass of liquid m₁ can be expressed as a function of the measuredpressure differential DP_(mes).

For preference, according to one possible advantageous feature of theinvention, the calculated liquid level h_(l) is corrected taking accountof an additional pressure difference value DP_(pipe) created by the gaspresent in the measurement pipes 11, 12, both when the pipes 11 aresituated inside the tank (FIG. 2) and outside the tank (FIG. 1).

What that means to say is that the pressure sensors 4 are remote and“read” pressures that are influenced by the fluid in the pipework 11, 12connecting them to the top and bottom parts of the tank.

The pressure differential DP_(mes) measured remotely between the top andbottom parts of the tank being connected to the so-called “real”pressure differential DP_(real) between the top and bottom parts of thetank according to the formula:

DP _(mes) =DP _(real) −DP _(pipe)

Scenario in which the Piping is Outside the Wall of the Tank (FIG. 1):

DP_(wall) is the pressure differential between the two ends of thevertical pipework running through the space between the walls (at thetop or at the bottom).

DP_(tot) _(—) _(length) is the pressure difference due to the pressureof gas in the part of pipework 11 connecting the uppermost point to theremote measurement member 4 (sensor).

DP_(amb) is the pressure difference due to the pressure of gas in thepart of pipework 11 connecting the lowermost point to the remotemeasurement member 4 (sensor).

The pressure differential DP_(wall) between the two ends of the verticalpiping passing through the space between the walls (at the top or at thebottom) can be considered to be substantially identical at the top andat the bottom (only the fact of gas in the pipework). Considering theshape of the lower pipework 12 in the space between the walls: thepipework runs close to the outer barrel to “pick up” heat energyexternal to the tank and completely vaporize the fluid in themeasurement pipework 12. Between the upper and lower ends of thisportion, the pressure is substantially the same (with a differential of0.5 bar at most).

Scenario in which the Pipework is in the Space Between the Walls (FIG.2):

DP_(side) _(—) _(gas) is the pressure difference in the part of the pipeconnected to the top part of the tank and on the gas side of the tank(containing gas), DP_(side) ₁₃ _(liq) is the pressure difference in thepart of the upper pipe lying on the liquid side of the tank (containingliquid).

The total mass m_(tot) of fluid in the tank (liquid and gas) can beexpressed as a function of data including, notably:

-   -   the measured pressure differential DP_(mes) between the top and        bottom parts of the tank (in Pa),    -   the density of the liquid in the tank ρ_(l),    -   the density of the gas in the tank ρ_(g),    -   the density of the gas in the pipe 11 on the gas side of the        tank measuring the pressure in the top part of the tank ρ_(side)        _(—) _(gas),    -   the density of the gas in the pipe 11 on the liquid side of the        tank measuring the pressure in the top part of the tank ρ_(side)        _(—) _(liquid),    -   the acceleration due to gravity of the Earth g,    -   the radius R of the tank,    -   the height of the elliptical part F,    -   the total height of the tank h_(tot)    -   the total mass m_(tot) of fluid in the tank can be expressed in        the form of an equation of the type given below, which will be        justified in greater detail hereinafter:

$\begin{matrix}{m_{tot} = {\pi \; {R^{2}\begin{bmatrix}{{\frac{{DP}_{mes}}{g}\left( \frac{\rho_{l -}\rho_{g}}{\begin{matrix}{{\rho_{l -}\rho_{g}} +} \\{\rho_{side\_ gas} - \rho_{side\_ liquid}}\end{matrix}} \right)} -} \\{{\frac{F}{3}\left( {\rho_{l +}\rho_{g}} \right)} + {\left( \frac{\begin{matrix}{\rho_{side\_ gas}\rho_{l -}} \\{\rho_{side\_ liquid}\rho_{g}}\end{matrix}}{\begin{matrix}{{\rho_{l -}\rho_{g}} + \rho_{side\_ gas} -} \\\rho_{side\_ liquid}\end{matrix}} \right)h_{tot}}}\end{bmatrix}}}} & \left( {{equation}\mspace{14mu} 101} \right)\end{matrix}$

This equation can be applied both before and after a filling of thetank.

It can be assumed that the densities of the gas and of the liquid areconstant before filling and after filling and equal to their meanvalues. Thus, by applying this formula 101 to the states beforem_(tot)(before) and after m_(tot)(after) filling, the mass of liquiddelivered m_(delivered) can be expressed asm_(delivered)=m_(tot)(after)−m_(tot)(before) and the geometric unknownsF and h_(tot) thus eliminated. In this way, the mass delivered (intheory known at the time of the delivery) can be expressed solely as afunction of the pressure differentials DP_(mes) measured before andafter filling DP_(mes) _(—) _(before) and DP_(mes) _(—) _(after) and ofthe radius R (which is unknown).

Thus, the radius R can be expressed solely as a function of the massdelivered, of the densities of the gas and of the liquid and of thepressure differentials, in the following form (equation 102):

$R = \sqrt{\frac{m_{delivered}{g\left\lbrack {1 - \left( \frac{\rho_{side\_ liquid} - \rho_{side\_ gas}}{\rho_{l -}\rho_{g}} \right)} \right\rbrack}}{\pi \left( {{DP}_{mes\_ after} - {DP}_{mes\_ before}} \right)}}$

However, it should be pointed out that, in practice, the operators'delivery notes are not always reliable in terms of the mass of liquidactually delivered m_(delivered). Specifically, errors may be due toincorrect transcription by the operator and/or losses of liquid duringhandling. Thus, any inaccuracy in the delivered mass m_(delivered) mayintroduce error into the estimate of the radius R of the tank. Toaddress this problem, one particular method described hereinbelow can beused in order to use only the reliable values of delivered massm_(delivered).

If the densities of the gas ρ_(g) and of the liquid ρ_(l) are consideredto be constant before and after filling and equal to their mean values,then the ratio r between, on the one hand, the mass delivered and, onthe other hand, the difference between the pressure differentialsDP_(mes) measured before and after filling (DP_(mes) _(—)_(after)−DP_(mes) _(—) _(before)) is constant.

Put differently (equation 103):

$r = {\frac{m_{delivered}}{{DP}_{mes\_ after} - {DP}_{mes\_ before}} = {{constant}\mspace{14mu} \left( {{in}\mspace{14mu} {{kg}/{Pa}}} \right)}}$

Thus, in order to select the reliable delivered masses, for eachdelivery note the method may:

-   -   1) calculate this ratio for each filling    -   2) calculate the relative difference between this ratio r and        the mean value r_(mean) of this ratio    -   3) select those delivery notes (delivered masses) which, in        terms of absolute value, do not deviate excessively from the        mean r_(mean) (for example do not exhibit more than 10%        divergence).

For preference, only these data are used for calculating the radius(equation 102).

The density of the liquid in the tank ρ_(l) is considered to be equal tothe mean of the densities of the liquid before ρ_(l) _(—) _(before) andafter ρ_(l) _(—) _(after) filling (equation 104):

$\rho_{l} = \frac{\rho_{l\_ before} + \rho_{l\_ after}}{2}$

It is also assumed that the density of the liquid before filling ρ_(l)_(—) _(before) is equal to the density at equilibrium at the pressure ofthe tank (equation 105):

ρ_(l) _(—) _(before)=ρ_(l) _(—) _(eq) _(—) _(tank)

It is assumed that the density of the liquid after filling ρ_(l) _(—)_(after) is equal to the mean of, on the one hand, the density of theliquid at equilibrium at the pressure of the tank before fillingweighted by the fraction of the volume occupied by this liquid and, onthe other hand, the density of the liquid in the truck considered atequilibrium at the pressure of the truck weighted by the fraction of thevolume available in the tank before filling. This leads to (equation106):

ρ_(l) _(—) _(after)=0.7*ρ_(l) _(—) _(eq) _(—) _(truck)+0.3*ρ_(l) _(—)_(eq) _(—) _(tank)

The density of the gas in the tank ρ_(g) is calculated at the pressureof the tank ρ_(tank) and for a temperature 20 K higher than theequilibrium temperature. Specifically, the gas in the tank is heated upafter filling by comparison with its equilibrium temperature (isapproximately 40 K above equilibrium) just before the next filling. The20 K value is a mean that may advantageously be chosen.

This then yields the next expression (equation 107 which gives thedensity of the gas as calculated as a function of a number ofparameters):

ρ_(g)=ρ_(g)(T _(g) =T _(eq) _(—) _(tank)+20K,P _(tank))

The density ρ_(side) _(—) _(gas) of the gas in the pipework 11connecting the top part of the tank and situated inside the tank (in thespace between the walls), that is to say the gas situated in thepipework on the gas side of the tank can be calculated at the pressureof the tank and at a temperature T_(gg) given by the following formula(equation 108):

$\quad\left\{ \begin{matrix}{T_{gg} = {T_{g} + {\frac{d\_ pipe}{w\_ length}\left( {T_{amb} - T_{g}} \right)}}} & \begin{matrix}{{{if}\mspace{14mu} {the}\mspace{14mu} {pipework}\mspace{14mu} {is}\mspace{14mu} {on}\mspace{14mu} {the}}\;} \\{outside}\end{matrix} \\{T_{gg} = T_{amb}} & \begin{matrix}{{{if}\mspace{14mu} {the}\mspace{14mu} {pipework}\mspace{14mu} {is}\mspace{14mu} {on}\mspace{14mu} {the}}\;} \\{inside}\end{matrix}\end{matrix} \right.$

Where:

-   -   d_pipe=the distance (spacing) between the upper pipework 11 and        the wall of the interior tank,    -   w_length=the thickness of the insulation of the internal tank,        and    -   T_(amb)=the ambient temperature around the tank.

In the case of pipework 11 situated inside (in the space between thewalls), it is possible to consider a linear temperature profile throughthe thickness of the insulation. In the case of external pipework 11,the temperature of the gas in the pipework 11 is considered to be equalto that of ambient temperature.

The density ρ_(side) _(—) _(liquid) of the gas in the pipework 11 on theside of the liquid phase in the tank is calculated at the currentpressure of the tank and at a temperature T_(gl) using the followingrelationship (equation 109):

$\quad\left\{ \begin{matrix}{T_{gl} = {T_{l} + {\frac{d\_ pipe}{w\_ length}\left( {T_{amb} - T_{l}} \right)}}} & \begin{matrix}{{{if}\mspace{14mu} {the}\mspace{14mu} {pipework}\mspace{14mu} {is}\mspace{14mu} {in}\mspace{14mu} {the}}\;} \\{{space}\mspace{14mu} {between}\mspace{14mu} {the}\mspace{14mu} {walls}}\end{matrix} \\{T_{gl} = T_{amb}} & \begin{matrix}{{{if}\mspace{14mu} {the}\mspace{14mu} {pipework}\mspace{14mu} {is}\mspace{14mu} {outside}\mspace{14mu} {the}}\;} \\{walls}\end{matrix}\end{matrix} \right.$

Likewise, for pipework 11 inside the tank on the same side as the liquidphase contained in the tank, consideration is given to a temperatureprofile that is linear between the liquid situated in the tank and theambient temperature on the outside around the tank.

The volume of the liquid V_(l) in a cylindrical tank of radius R andhaving an elliptical end of height F is given by the relationship(equation 110):

$\left. {{{If}\mspace{14mu} h_{l}} \geq F}\Rightarrow V_{l} \right. = {\pi \; {R^{2}\left\lbrack {h_{l} - \frac{F}{3}} \right\rbrack}}$$\left. {{{If}\mspace{14mu} h_{l}} < F}\Rightarrow V_{l} \right. = {{\frac{2}{3}\pi \; {FR}^{2}} - {{\pi \left( {F - h_{l}} \right)}\left\lbrack {R_{2} - {\frac{R^{2}}{3F^{2}}\left( {F - h_{l\;}} \right)^{2}}} \right\rbrack}}$

(where h_(l)=the height of liquid in the tank).

If the total height of the tank is defined as h_(tot), then we can write(equation 111):

h _(tot) =H _(max) +ov_length

Where ov_length=the minimum height of gas in the tank from the top endthereof.

The volume of gas V_(g) in the tank is the complement of the volume ofliquid V_(l), with respect to the total volume of the tank V_(tot)according to the relationship (equation 112):

$V_{tot} = {\pi \; {R^{2}\left( {h_{tot} - \frac{2F}{3}} \right\rbrack}}$V_(g) = V_(tot) − V_(l)

The mass of fluid in the tank is equal to the sum of the liquid and ofthe gas (equation 113):

m _(tot)=ρ_(l) V _(l)+ρ_(g) V _(g)

When h_(l) is greater than or equal to F (which it is most of the time),using equations 110 and 113, the formula expressing mass can besimplified to give (equation 114):

$m_{tot} = {\pi \; {R^{2}\begin{bmatrix}{\frac{{DP}_{mes}}{g} - {\frac{F}{3}\left( {\rho_{l} + \rho_{g}} \right)} +} \\{{\left( {\rho_{side\_ liquid} - \rho_{side\_ gas}} \right)h_{l}} + {\rho_{side\_ gas}h_{tot}}}\end{bmatrix}}}$

The height of liquid in the tank is therefore linked to the differentialpressure measurement DP_(mes) according to the following relationship(equation 115):

$h_{l} = \frac{\frac{{DP}_{mes}}{g} - {\left( {\rho_{g} - \rho_{side\_ gas}} \right)h_{tot}}}{\rho_{l} - \rho_{g} + \rho_{side\_ gas} - \rho_{side\_ liquid}}$

Thus, equations 114 and 115 lead to equation 101 given hereinabove.

The radius R of the tank can thus be calculated and estimated for eachplausible value of mass delivered during a filling operation. The meanradius can thus be calculated on the basis of these multiplecalculations. This is the first parameter determined from just measuringthe pressure differential DP, the mass of liquid delivered, and a fewapproximations regarding densities.

Calculating the Height F of the Elliptical End Part

This end height F can be deduced directly from the estimated radius R,as the radio between these two geometric parameters is considered to besubstantially constant across all tank manufacturers. This secondgeometric parameter can be deduced (cf. equations 116 hereinbelow fortwo examples of manufacturer).

$\frac{R}{F} = {1.9\mspace{14mu} {for}\mspace{14mu} {tanks}\mspace{14mu} {made}\mspace{14mu} {by}\mspace{14mu} {``{Cryolor}"}}$$\frac{R}{F} = {2\mspace{14mu} {for}\mspace{14mu} {tanks}\mspace{14mu} {made}\mspace{14mu} {by}\mspace{14mu} {``{Chart}"}}$

Where the manufacturer is unknown, the approximation of 1.95 can be usedfor example.

Estimating the Maximum Height of Liquid and the Total Height (H_(max)and h_(tot)).

For most cryogenic tanks, the ratio between the maximum volume of liquidV_(max) and the total volume of liquid V_(tot) is constant and dependentonly on the level of pressure in tank. This ratio is 0.95 for tanks atlow and medium pressure (ranging between 1 and 15 bar) and is 0.90 forhigh pressures (in excess of 15 bar), (cf. equation 117):

$\frac{V_{\max}}{V_{tot}} = {0.95\mspace{14mu} {for}\mspace{14mu} {tanks}\mspace{14mu} {at}\mspace{14mu} {low}\mspace{14mu} {and}\mspace{14mu} {medium}\mspace{14mu} {pressure}}$$\frac{V_{\max}}{V_{tot}} = {0.90\mspace{14mu} {for}\mspace{14mu} {tanks}\mspace{14mu} {at}\mspace{14mu} {high}\mspace{14mu} {pressure}}$

In the knowledge that (equation 118):

$V_{tot} = {\pi \; {R^{2}\left\lbrack {h_{tot} - \frac{2F}{3}} \right\rbrack}}$$V_{\max} = {\pi \; {R^{2}\left\lbrack {H_{\max} - \frac{F}{3}} \right\rbrack}}$

We obtain (equation 119)

$H_{\max} = {\frac{F}{3} + {\frac{V_{\max}}{V_{tot}}\left\lbrack {h_{tot} - \frac{2F}{3}} \right\rbrack}}$

Thus, once the total height h_(tot) is determined the maximum height ofliquid H_(max) can be deduced using this last equation (knowing

$\left. \frac{V_{\max}}{V_{tot}} \right)$

in order thereafter to estimate F.

The total height of the tank is determined and estimated from thepressure differential measured just after a filling DP_(mes) _(—)_(after) making the assumption that filling is total. In such an event,the measured pressure differential can be expressed in the form(equation 120):

DP _(mes) _(—) _(after)=ρ_(l) _(—) _(after) H _(max) g+ρ _(g) _(—)_(after)(h _(tot) −H _(max))g−(ρ_(gg) _(—) _(before) h _(g) _(—)_(before)+ρ_(gl) _(—) _(before) h _(l) _(—) _(before))g

In this equation 120, the densities of gas in the measurement pipework11 ρ_(side) _(—) _(gas) _(—) _(before) and ρ_(side) _(—) _(liquid) _(—)_(before) are calculated at the pressure of the tank after filling butwith pre-filling temperatures for the gas in the pipework 11 (equation121):

ρ_(side) _(—) _(gas) _(—) _(before)=ρ_(g)(T _(gg) _(—) _(before) ,P_(tank) _(—) _(after))

ρ_(side) _(—) _(liquid) _(—) _(before)=ρ_(g)(T _(gl) _(—) _(before) ,P_(tank) _(—) _(after))

This is the result of the thermal inertia of the tank insulation lyingin the space between the walls. In actual fact, the characteristic timefor the conduction of heat through this thickness of insulation (0.045 mof perlite for example) can be calculated using the following equation(equation 122):

$\tau = {\frac{e^{2}}{a} = {\frac{0.045^{2}}{8.6 \times 10^{- 7}} = {0.66\mspace{14mu} {hour}}}}$

Where e=the thickness of insulation and a=the thermal diffusivity of theinsulation.

The time required for thermal stability of the gas in the pipework 11lying inside is at least twice the duration τ (of the order of about1.33 hours), which is greater than the mean filling time (which is about0.4 hours).

The height of liquid in the tank before filling is estimated at 30% ofthe maximum height of liquid and the height of gas can be deducedtherefrom thereafter (equations 123):

h_(l) _(—) _(before)=0.3H_(max)

h _(g) _(—) _(before) =h _(tot)−0.3H _(max)

Detecting Complete Fillings

In order to detect whether a filling is complete from the list of datacovering a plurality of fillings, it is possible to use the followingprocedure making the assumption that at least one filling in the list offillings for which data is available is a complete filling.

1) The maximum pressure differential DP just after a filling and for allfillings is determined.

2) A filling is considered to be a complete or total filling if therelative difference between the pressure differential just after fillingand the maximum pressure differential is below a threshold value (forexample 5%).

Preferably, only fillings considered to be complete are used fordetermining the total height of the tank. It must be emphasized that thegreater the volume of the tank, the greater the probability ofincomplete filling. This can be explained by the fact that the volumesof truck deliveries are limited by the storage capacity of the truck. Asa result, the greater the volume that is to be filled, the morenecessary it will be to have a great deal of filling data available.

Total Height of the Tank and Estimating the Maximum Liquid Level(h_(tot) and H_(max)).

For each filling considered to be complete, the total height isdetermined using equation 120. Next, the mean value is calculated andthis provides the third geometric parameter of the tank.

The maximum height of liquid H_(max) is itself deduced from the totalheight of the tank using equation 119. That gives a fourth geometricparameter.

For European tanks with a volume of 50 m³ or greater, filling is rarelycomplete. In such cases, the total height of the tank can be determinedon the basis of an overall estimate using the following equation 124which is based on the previous equation 11.

$h_{tot} = {\frac{V_{tot}}{\pi \; R^{2}} + \frac{2F}{3}}$

Thus, the maximum height of liquid H_(max) can be deduced from equation119.

Estimating Thermal Losses

The thermal losses of the cryogenic tank are generally expressed aspercent oxygen lost per day. According to the invention, it would appearto be sufficient to make an overall estimate of this loss (thisparameter is less sensitive or important than the total height).

This loss is a function of the type of volume of the tank, and decreasesas the volume increases. For example, for a Cryolor tank, a correctapproximation is (equation 125)

% O₂ lost per day=0.65273V_(tot) ^(−0.37149)

Knowing the volume of the tank in m³, this last equation estimates dailythermal losses which is the fifth parameter of the tank. This equationcan be used for other types of tank (by other manufacturers).

The method described hereinabove makes it possible to estimate, withgood precision, the geometric parameters and thermal loss parametersusing simple measurements of pressure differentials, pressures and massdelivered at the time of deliveries.

1-10. (canceled)
 11. A method of estimating characteristic geometricparameters of a double-walled cryogenic tank that is insulated with orwithout vacuum comprising: a step of measuring the pressure differential(in Pa) between the top and bottom parts of the tank before a filling(DP_(mes) _(—) _(before)); a step of measuring the pressure differential(in Pa) between the top and bottom parts of the tank after said filling(DP_(mes) _(—) _(after)); a step of determining the mass of liquiddelivered (m_(delivered)) (in kg) during said filling; and a step ofcalculating a first geometric parameter (R) of the tank, the tankcomprising a cylindrical portion and at least one end that has anelliptical portion of set height (F), wherein the first geometricparameter is the radius (R) of the cylinder (in m), said first geometricparameter (R) being calculated from: the mass of liquid delivered(m_(delivered)) (in kg) determined during the filling; the differencebetween the pressure differentials (DP_(mes)) measured before and afterfilling (DP_(mes) _(—) _(after)−DP_(mes) _(—) _(before)); and thedensities of the gas and of the liquid (ρ_(g), ρ_(l)) in the tank, theradius (R) (in m) is calculated using the equation:$R = \sqrt{\frac{m_{delivered}{g\left\lbrack {1 - ({MAVO})} \right\rbrack}}{\pi \left( {{DP}_{mes\_ after} - {DP}_{mes\_ before}} \right)}}$in which the pressure differentials are expressed in Pa, π is the numberPi, g is the acceleration due to gravity of the Earth (in m/s²) and MAVOis a dimensionless corrective coefficient that is a function of thedensity.
 12. The method of claim 11, wherein, in order to estimate thedensity coefficient (MAVO), the method uses at least one of thefollowing assumptions: the density of the liquid in the tank ρ_(l) isconsidered to be equal to the mean of the densities of the liquid beforeρ_(l) _(—) _(before) and after ρ_(l) _(—) _(after) filling, the densityof the liquid before filling ρ_(l) _(—) _(before) is equal to thedensity at equilibrium at the pressure of the tank, the density of theliquid after filling ρ_(l) _(—) _(after) is equal to the mean of, on theone hand, the density of the liquid at equilibrium at the pressure ofthe tank before filling weighted by the fraction of the volume occupiedby this liquid and, on the other hand, the density of the liquid in thetruck considered at equilibrium at the pressure of the truck, weightedby the fraction of the volume available in the tank before filling, thevolume of the liquid in the tank before filling is considered to beequal to a known fraction (for example 30%) of the maximum volume ofliquid in the tank, the density of the gas in the tank ρ_(g) iscalculated at the pressure of the tank P_(tank) and for a temperaturethat is increased over the equilibrium temperature at the pressure ofthe tank (for example increased by 20 K), the measured pressuredifferential DP_(mes) is corrected to take account of an additionalpressure different value (DP_(pipe)) created by the gas present in themeasurement pipes in the event of remote measurement.
 13. The method ofclaim 11, wherein the volume of the liquid V_(l) in a cylindrical tankof radius R having one elliptical end of height F is given by therelationship:$\left. {{{If}\mspace{14mu} h_{l}} \geq F}\Rightarrow V_{l} \right. = {\pi \; {R^{2}\left\lbrack {h_{l} - \frac{F}{3\;}} \right\rbrack}}$$\left. {{{If}\mspace{14mu} h_{l}} > F}\Rightarrow V_{l} \right. = {{2\pi \; {FR}^{2}} - {{\pi \left( {F - h_{l}} \right)}\left\lbrack {R^{2} - {\frac{R^{2}}{3F^{2}}\left( {F - h_{l}} \right)^{2}}} \right\rbrack}}$h_(l) being the height of liquid in the tank.
 14. The method of claim11, wherein the method comprises a step of calculating a secondgeometric parameter consisting of the height (F) of the ellipticalportion of the tank as a function of calculated value of radius (R), thevalue of the height (F) of the elliptical portion being given by thefollowing equation: $\frac{R}{F} = K$ K being a known constantrepresentative of a type of tank or an arbitrarily chosen constant suchas about 1.95.
 15. The method of claim 11, wherein the method comprisesa step of calculating a third geometric parameter consisting of thetotal height of the tank (h_(tot)) from the pressure differentialmeasured just after a filling DP_(mes) _(—) _(after) assuming thatfilling is total, this measured pressure differential being expressed inthe following form:DP _(mes) _(—) _(after) =A ₀ H _(max) +A ₁ h _(tot) +A ₂ h _(g) _(—)_(before) +A ₃ h _(l) _(—) _(before) in which A₀, A₁, A₂, A₃ arecoefficients dependent on the densities of the gas and of the liquidbefore and after filling, H_(max) is the maximum height of liquid in thetank, h_(g) _(—) _(before) being the height of gas in the tank beforefilling, h_(l) _(—) _(before) being the height of liquid in the tankbefore filling, and using the following assumption: the height of liquidin the tank before filling h_(l) _(—) _(before) is estimated at a knownset threshold FS expressed as a percentage of the maximum height ofliquid H_(max), the height h_(g) _(—) _(before) of gas before fillingbeing deduced therefrom as being the complement:h_(l) _(—) _(before)=FS H_(max)h_(g) _(—) _(before)=h_(tot−FS H) _(max)
 16. The method of claim 15,wherein the method comprises a step of calculating a fourth geometricparameter consisting of the maximum height of liquid H_(max), this beingis deduced the total height of the tank calculated from the followingequation:$H_{\max} = {\frac{F}{3} + {\frac{V_{\max}}{V_{tot}}\left\lbrack {h_{tot} - \frac{2F}{3}} \right\rbrack}}$17. The method of claim 11, wherein the method comprises a step ofcalculating a fifth geometric parameter consisting of the thermal lossof the tank, said thermal loss expressed as an oxygen percentage (% O₂)of oxygen lost per day being approximated using a relationship of thetype:% O₂ lost per day=p_(l)V_(tot) ^(−p) ² in which V_(tot) is the totalvolume of the tank, p_(l) is a coefficient of the order of 0.6 andpreferably equal to 0.65273 and O₂ is a coefficient of the order of 0.3and preferably equal to 0.37149.
 18. The method of claim 11, wherein themethod comprises a step of collecting a plurality of values for the massof liquid delivered (m_(delivered)) which are determined respectivelyduring a plurality of fillings of the tank, a step of calculating, foreach value of mass of liquid delivered (m_(delivered))₅ the fill ratio r$r = {\frac{m_{delivered}}{{DP}_{mes\_ after} - {DP}_{mes\_ before}} = {constant}}$and in that the method of estimating uses only those values of mass ofliquid delivered (m_(delivered)) for which the absolute value of thefill ratio r differs with respect to a set constant reference value byno more than a fixed threshold amount.
 19. The method of claim 18,wherein the set constant reference value for the fill ratio r consistsof the mean of this fill ratio r calculated for a plurality of fillings.20. The method of claim 11, wherein when the measured pressuredifferentials (DP) do not correspond to the real pressure differentials,that is to say when the pressure is measured remotely via measurementpipes (11, 12) situated inside the tank in the space between the walls,thus creating an additional pressure difference, the coefficient MAVO isgiven by the formula${MAVO} = \frac{\rho_{side\_ liquid} - \rho_{side\_ gas}}{\rho_{l} - \rho_{g}}$wherein: ρ_(l)=the density of the liquid in the tank, ρ_(g)=the densityof the gas in the tank, ρ_(side) _(—) _(gas)=the density of the gas inthe pipe on the gas side of the tank measuring the pressure in the toppart of the tank, ρ_(side) _(—) _(liquid)=the density of the gas in thepipe on the liquid side of the tank measuring the pressure in the toppart of the tank, and in that when the measured pressure differentialsDP do correspond to the real pressure differentials (e.g.: when thepressures are measured remotely via measurement pipes situated on theoutside of the tank and at ambient temperature ρ_(side) _(—)_(liquid)=ρ_(side) _(—) _(gas)), the coefficient MAVO=0 (zero).
 21. Themethod of claim 12, wherein the volume of the liquid V_(l) in acylindrical tank of radius R having one elliptical end of height F isgiven by the relationship:$\left. {{{If}\mspace{14mu} h_{l}} \geq F}\Rightarrow V_{l} \right. = {\pi \; {R^{2}\left\lbrack {h_{l} - \frac{F}{3\;}} \right\rbrack}}$$\left. {{{If}\mspace{14mu} h_{l}} < F}\Rightarrow V_{l} \right. = {{\frac{2}{3}\pi \; {FR}^{2}} - {{\pi \left( {F - h_{l}} \right)}\left\lbrack {R^{2} - {\frac{R^{2}}{3F^{2}}\left( {F - h_{l}} \right)^{2}}} \right\rbrack}}$h_(l) being the height of liquid in the tank.
 22. The method of claim13, wherein the method comprises a step of calculating a secondgeometric parameter consisting of the height (F) of the ellipticalportion of the tank as a function of calculated value of radius (R), thevalue of the height (F) of the elliptical portion being given by thefollowing equation: $\frac{R}{F} = K$ K being a known constantrepresentative of a type of tank or an arbitrarily chosen constant suchas about 1.95.
 23. The method of claim 14, wherein the method comprisesa step of calculating a third geometric parameter consisting of thetotal height of the tank (h_(tot)) from the pressure differentialmeasured just after a filling DP_(mes) _(—) _(after) assuming thatfilling is total, this measured pressure differential being expressed inthe following form:DP _(mes) _(—) _(after) =A ₀ H _(max) +A ₁ h _(tot) +A ₂ h _(g) _(—)_(before) +A ₃ h _(l) _(—) _(before) in which A₀, A₁, A₂, A₃ arecoefficients dependent on the densities of the gas and of the liquidbefore and after filling, H_(max) is the maximum height of liquid in thetank, h_(g) _(—) _(before) being the height of gas in the tank beforefilling, h_(l) _(—) _(before) being the height of liquid in the tankbefore filling, and using the following assumption: the height of liquidin the tank before filling h_(l) _(—) _(before) is estimated at a knownset threshold FS expressed as a percentage of the maximum height ofliquid H_(max), the height h_(g) _(—) _(before) of gas before fillingbeing deduced therefrom as being the complement:h_(l) _(—) _(before)=FS H_(max)h_(g) _(—) _(before)=h_(tot−)FS H_(max)
 24. The method of claim 16,wherein the method comprises a step of calculating a fifth geometricparameter consisting of the thermal loss of the tank, said thermal lossexpressed as an oxygen percentage (% O₂) of oxygen lost per day beingapproximated using a relationship of the type:% O₂ lost per day=p_(l)V_(tot) ^(−p) ² in which V_(tot) is the totalvolume of the tank, p_(l) is a coefficient of the order of 0.6 andpreferably equal to 0.65273 and p₂ is a coefficient of the order of 0.3and preferably equal to 0.37149.
 25. The method of claim 17, wherein themethod comprises a step of collecting a plurality of values for the massof liquid delivered (m_(delivered)) which are determined respectivelyduring a plurality of fillings of the tank, a step of calculating, foreach value of mass of liquid delivered (m_(delivered)), the fill ratio r$r = {\frac{m_{delivered}}{{DP}_{mes\_ after} - {DP}_{mes\_ before}} = {constant}}$and in that the method of estimating uses only those values of mass ofliquid delivered (m_(delivered)) for which the absolute value of thefill ratio r differs with respect to a set constant reference value byno more than a fixed threshold amount.
 26. The method of claim 19,wherein when the measured pressure differentials (DP) do not correspondto the real pressure differentials, that is to say when the pressure ismeasured remotely via measurement pipes (11, 12) situated inside thetank in the space between the walls, thus creating an additionalpressure difference, the coefficient MAVO is given by the formula${MAVO} = \frac{\rho_{side\_ liquid} - \rho_{side\_ gas}}{\rho_{l} - \rho_{g}}$wherein: ρ_(l)=the density of the liquid in the tank, ρ_(g)=the densityof the gas in the tank, ρ_(side) _(—) _(gas)=the density of the gas inthe pipe on the gas side of the tank measuring the pressure in the toppart of the tank, ρ_(side) _(—) _(liquid)=the density of the gas in thepipe on the liquid side of the tank measuring the pressure in the toppart of the tank, and in that when the measured pressure differentialsDP do correspond to the real pressure differentials (e.g.: when thepressures are measured remotely via measurement pipes (11, 12) situatedon the outside of the tank and at ambient temperature ρ_(side) _(—)_(liquid)=ρ_(side) _(—) _(gas)), the coefficient MAVO=0 (zero).